On 21/04/2013 10:03 AM, Frederick Williams wrote:> fom wrote:>>>> On 4/20/2013 3:40 PM, Frederick Williams wrote:>>> Nam Nguyen wrote:>>>>>>>> On 20/04/2013 8:59 AM, fom wrote:>>>>> On 4/20/2013 5:25 AM, Alan Smaill wrote:>>>>>> Frederick Williams <freddywilliams@btinternet.com> writes:>>>>>>>>>>>>> Nam Nguyen wrote:>>>>>>>>>>>>>>>> On 19/04/2013 5:55 AM, Frederick Williams wrote:>>>>>>>>> Nam Nguyen wrote:>>>>>>>>>>>>>>>>>>>> On 18/04/2013 7:19 AM, Frederick Williams wrote:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Also, as I remarked elsewhere, "x e S' /\ Ay[ y e S' -> y e S]">>>>>>>>>>> doesn't>>>>>>>>>>> express "x is in a non-empty subset of S".>>>>>>>>>>>>>>>>>>>> Why?>>>>>>>>>>>>>>>>>> It says that x is in S' and S' is a subset of S.>>>>>>>>>>>>>>>> How does that contradict that it would express "x is in a non-empty>>>>>>>> subset of S", in this context where we'd borrow the expressibility>>>>>>>> of L(ZF) as much as we could, as I had alluded before?>>>>>>>>>>>>>> You really are plumbing the depths. To express that x is non-empty you>>>>>>> have to say that something is in x, not that x is in something.>>>>>>>>>>>> but the claim was that x *is in* a non-empty set -->>>>>> in this case S', which is non-empty, since x is an element of S',>>>>>> and S' is a subset of S.>>>>>>>>>>>> (Much though it would be good for Nam to realise that>>>>>> some background set theory axioms would be kind of useful here)>>>>>>>>>>>>>>>> Yes. I thought about posting some links indicating>>>>> that primitive symbols are undefined outside of a>>>>> system of axioms (definition-in-use)>>>>>>>>>> The other aspect, though, is that Nam appears to be using an>>>>> implicit existence assumption. So,>>>>>>>>>> AxASES'(xeS' /\ Ay(yeS' -> yeS))>>>>>>>>>> clarifies the statement and exhibits its second-order nature.>>>>> This is fine since he claims that his work is not in the>>>>> object language.>>>>>>>> Right.>>>>>> If fom's formula is to express "x is in a non-empty subset of S" then it>>> needs to have both x and S free, so delete the first two quantifiers.>>>>>>> Do you have a particular x and S in mind?>> I probably misunderstood. If Nam saying that, for every x and every set> S, x is in a non-empty subset of S, then your formula expresses that.> But clearly it is false.

I didn't say "every x and every set S". In the underlying context I wastalking about, x is _an_ individual and S is _a_ set (however generaleach might be).

>>> Or are we reverting to the distinction between real>> and apparent variables from the first "Principia>> Mathematica"?>> I call them free and bound respectively.>>> Or are we interpreting a statement in relation to a>> general usage over an unspecified domain? My quantifiers>> are in place to make clear the meaning for general usage.>>>> Within any context involving proof, the leading quantifiers>> obey rules:>>>> |AxASES'(xeS' /\ Ay(yeS' -> yeS))>> |ASES'(teS' /\ Ay(yeS' -> yeS))>> |ES'(teS' /\ Ay(yeS' -> yeP))>> ||(xeP' /\ Ay(yeP' -> yeP))>>>> The original statement is assumed (hence, is stroked)>>>> The existential statement is assumed (hence, a second stroke)>>>> Now the presuppostions of use are clear.>>>> That was my only purpose.

-- ----------------------------------------------------There is no remainder in the mathematics of infinity.